1. Introduction

Rotating beams have attracted great interest for more than three decades due to their properties present novel opportunities for scientific research and technological applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For example, the ability of rotating beams to rotate objects offers a new control degree for micro-objects and has important applications in optical micromachines and biotechnology. All such applications require a comprehensive understanding of the interference patterns, transverse energy flow, and angular momentum within rotating beams. Besides the use of specially fabricated micro-objects [12], two major schemes of rotating beams have successfully implemented. The first is Laguerre-Gaussian (LG) beams [13,14,15], which have an on-axis phase singularity and are characterized by helical phase fronts. The Poynting vector in such a beam follows a corkscrew-like path as the beam propagates, giving rise to transverse energy flow in the beam. The other model is the interference pattern rotation. Specifically, an interferometric pattern can be between an LG beam and a plane wave [16], or a LG beam and a Gaussian beam [17], or two annular laser beams [18,19]. The interference pattern can be made to rotate around the axis of the beam by displacing the two beams either through a change in the relative azimuthal index of the two beams or by creating a frequency difference between the two beams.

Interaction between light and material is the key to exploiting and controlling the property of light. From this simple observation, we might provide new possibilities for steering the rotation of rotating beam by utilizing left-handed materials (LHMs), namely, the materials that have simultaneously negative electric susceptibility and magnetic permeability [20]. The change in sign of the refractive index of the material could lead to some unique phenomena in Gaussian beams, such as inverse Gouy phase shift and the negative Rayleigh length [21]. While the spiral of the Poynting vector of a LG beam was found to be proportional to the Gouy-phase shift [22, 23]. Hence, it is interesting to investigate the propagation characteristics of rotating beams in LHMs. More recently, the interesting effect of reversed wavefront associated with a monochromatic LG beam in LHMs was revealed [24], and the rotational Doppler effect in LHMs was found to be unreversed, significantly different from the linear Doppler effect [25]. As we know, a realistic LHM must be dispersive, which is necessary for a causal medium with negative index of refraction. Accordingly, it is expected that the dispersion effect for non-monochromatic rotating beams will become significant. An interesting question naturally arises: how the propagation characteristics of such beams are influenced by the negative refraction index as well as the dispersive property of LHMs? In the present paper, we try to answer this question by introducing the rotating beams formed by superpositions of coaxial LG modes with different frequencies.

To disclose the properties of rotating beams in dispersive LHMs, first we use the representation of plane-wave angular spectrum to obtain the analytical description of a LG beam propagation in LHMs. Then, we attempt to study the rotation of interference patterns of rotating beams while propagating in LHMs and recover how the transverse Poynting vector evolve in LHMs. We will find a possibility of reversing the direction of transverse energy flow, with the direction of interference pattern not being affected. Finally, we focus our interest on the angular momentum density and the angular momentum flow of rotating beams. We will find that the angular momentum density may be parallel or antiparallel to the transverse energy flow when the rotating beam propagates in LHMs, while the angular momentum flow is always opposite to the transverse energy flow.

2. Paraxial propagation of a Laguerre-Gaussian beam in LHMs

Before investigating the role of dispersion in the propagation of rotating beams, first let us discuss the case of the paraxial propagation of LG beams in LHMs. For a comparison, the corresponding propagation characteristics in RHMs will also been discussed. Let’s start from the representation of plane-wave angular spectrum. From the point view of Fourier optics, we know that if the complex field distribution of a monochromatic disturbance is Fourier-analyzed across any plane, the various spatial Fourier components can be identified as plane waves traveling in different directions away from that plane. The field amplitude at any other point (or across any other parallel plane) can be calculated by adding the contributions of these plane wave, taking due account of the phase shifts they have undergone during propagation [26].

Suppose that, due to some monochromatic sources, a wave is incident on a transverse (x,y) plane traveling with a component of propagation in the positive z direction. Let the complex field across the z = 0 plane be represented by u(r,0), where r = (x,y) is the transverse radius-vector. The angular spectrum is related to the boundary distribution of the field by means of the relation

where k = n ω/c is wave number, n is the refractive index, ω is the angular frequency, c is the speed of light in vacuum, J_l is the first kind Bessel function with order l, r is polar coordinate. The two-dimensional Fourier transformations of Eq. (1) can be easily obtained from an integration table [27]. The function u(r,0) together with Eq. (1) provide the expression of the field in the space z > 0, which yields

Equation (2) is a standard two-dimensional Fourier transform [26], where z is the distance from the beam waist, n_R,L are the refractive index of RHM and LHM, respectively, k ₀ is the wave number in vacuum. The field u(r,z) is the slowly varying envelope amplitude.

Using the representation described above, let us consider a LG beam with frequency ω propagation in RHMs and LHMs, respectively. Without any loss of generality, we assume that the beam waist locates at the object plane z = 0, which is described by the complex amplitude distribution

where c_pl is weight coefficient, w ₀ is the beam waist radius, R ₀ is the beam waist radius of curvature of wavefront, L_p ^|l| is the generalized Laguerre polynomial. A given mode is usually denoted as LG_p^l, where l and p are the two integer indices that describe the mode: l refers to the number of 2π phase cycles around the circumference of the mode, so that l is known as the azimuthal index, and (p + 1) indicates the number of radial nodes in the mode profile. A LG beam is well known to possess orbital angular momentum due to an exp(ilφ) phase term (where φ is the azimuthal phase) in the mode description that gives rise to a well defined orbital angular momentum of lh̄ per photon. This orbital angular momentum lh̄ is distinct from the spin angular momentum due to the polarization state of the light [3]. According to Eqs. (1)-(3), the field of LG mode travels in RHMs can be written as

where $w\left(z\right)=w_0\sqrt{\left(z^2+z_R^2\right)/z_R^2}$ is the beam size, z_R = k ₀ n_R w ² ₀/2 is the corresponding Rayleigh range, R(z) = (z ² + z_R ²)/z is the radius of curvature of wavefront, and ψ(z) = arctan(z/z_R) is the Gouy phase. Similarly, the field of LG mode travels in LHMs can be written as

where $w\left(z\right)=w_0\sqrt{\left(z^2+z_L^2\right)/z_L^2},$ z_L = k ₀ n_L w ² ₀/2, R(z) = (z ² + z_L ²)/z and ψ(z) = arctan(z/z_L) are the beam size, the corresponding Rayleigh range, the radius of curvature of wavefront and the Gouy phase in LHMs, respectively. Because of the negative refractive index, the reversed Gouy-phase shift should be introduced. We have found that, the inverse Gouy-phase shift gives rise to an inverse spiral of Poynting vector [24]. Note that LG beams attenuate exponentially with distance, if n_L has a imaginary part. Consider the situation where the LG beam is linearly polarized. The electric and magnetic fields of a LG beam can be written as [28]

and

respectively, where e _x,e _y,e _z are the unit vectors of corresponding Cartesian axes, axis x is supposed to coincide with the direction of the transverse component electric field, and u is shorthand for u(r,z,t). These field expressions neglect terms in each component that are smaller than those retained in accordance with the paraxial approximation. The z components are smaller than the x and y components by a factor of order 1/kw ₀. It is readily verified that the fields satisfy Maxwell’s equations. Note that the Cartesian derivatives can be converted to polar r and φ derivatives in the usual way.

3. Rotating beams propagation in LHMs

Now we attempt to describe the evolution of a rotating beam in LHMs. The relative dielectric permittivity and magnetic permeability of the LHM are considered to be Lorentz model for the dispersive medium [29]:

where ω _e0 is the electronic resonance frequency, ω_ep is the electronic plasma frequency, γ_e is the electronic damping frequency, ω _m0 is the magnetic resonance frequency, ω_mp is the magnetic plasma frequency,and γ_m is the magnetic damping frequency.When ω _e0 = ω _m0 =0, the Lorentz model turns to the Drude model. Here, we assume that ω _e0 = ω _m0, ω_ep = ω_mp, and γ_e = γ_m. n is complex denoted by n(ω) = η(ω) + iκ(ω).

Consider such a rotating beam which is formed by superpositions of coaxial LG modes with different frequencies but opposite chirality(LG_p,+l + LG_p,-l). We assume that all modes have the equal magnitudes and waist parameters, both centered at axis z and paraxially propagating in the positive z direction. In the case of equal frequencies, such a combination is well known to be equivalent to an ordinary Hermite-Gaussian beam. For the brevity, we will call this beam a “rotating Hermite-Gaussian” (RHG) beam, which has an unique edge wavefront dislocation and a particular “multi-spot” intensity distribution within the transverse (x,y)-plane [8]. The electric and magnetic vectors of its field are

where indices ‘+’ and ‘-’ relate to ω ₊ = ω ₀ + Δω/2 and ω _- = ω ₀ - Δω/2 components, respectively. ω ₀ is the center frequency of RHG beam, Δω = (ω ₊ - ω _-) > 0. In their turn, the field vectors of a paraxial beam can be represented as

where k _± = n_R,L ω _±/c is the wave numbers. Note that, a physically consistent condition Δω ≪ ω _± is satisfied.

It is usual to consider the energy flow in an electromagnetic field in term of the Poynting’s theorem [30]. The theorem leads to a continuity equation or conservation law where the energy flow is represented by the Poynting vector:

which has the dimensions of energy per unit time per unit area. As usual, the total energy flow can be decomposed into transverse and longitudinal components

where S _⊥ = S _x + S _y. Discarding rapidly oscillating and thus unobservable terms, one can write

where

and c _± = 2p _± +|l _±|+1. We note that in the lossy LHM, k is complex denoted by k = τ + iυ. It is obvious from Eq. (15) that the longitudinal component of Poynting vector attenuates exponentially with distance. In particular, within the plane of analysis z = 0, Eq. (15) can be reduced to the form

Equation (17) determine the intensity pattern of Fig. 1, rotating with angular velocity Ω = Δω/(l ₊ -l _-). As usual, positive Ω corresponds to counter-clockwise rotation when seeing against the beam propagation axis (the right-screw rule). One can note that the properties of LHMs cannot reverse the transverse beam pattern rotation. In order to accurately describe this question, we investigate this problem with the help of model examples representing important features of RHG beams. Let us consider two typical patterns of the field, which are formed by a simple two-term superposition of the class LG_0,+1 and LG_0,-1, LG_1,-1 and LG_1,+1. In the case of LG_0,+1 + LG_0,-1, Ω > 0 and the intensity pattern exhibits anticlockwise spiral. In the another case of LG_1,-1 + LG_1,+1, Ω < 0 and the intensity pattern presents clockwise spiral.

4. Transverse energy flow and angular momentum density of rotating beams in LHMs

In this section, first we focus on the role of dispersion in the transverse energy flow of RHG beams propagating in the dispersive LHM. Similar to Eq. (15), S _x and S _y can be divided into the time-invariant and slowly oscillating parts.

 

Fig. 1. Evolution of the interference pattern of model spiral beams viewed against the beam propagation direction in the waist plane. The white dot indicates the rotation of the pattern due to the azimuthal index of two-term superposition of LG beams. Here, Ω is the angular velocity of interference pattern, T = 2(l ₊ - l _-)π/Δω is the period of rotation. (a)-(d) superposition LG_0,+1 +LG_0,-1: we know that Ω > 0 and the interference pattern exhibits anticlockwise rotation. (a’)-(d’) superposition LG_1,-1 +LG_1,+1: we get that Ω > 0 and the interference pattern exhibits clockwise rotation.

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One can note that the results constitute a pure contribution of the time-invariant part of the energy flow (the slowly varying part gives zero). The mechanical properties of a RHG beam find their convincing explanation in the detailed picture of the temporal behavior of field. In such a beam, the observable vortex-like beam structure is nothing else than the time-averaged pattern of this rotation [5]. The function of S _⊥ may be written in polar coordinates as

The results of Eqs. (21) and (22) prove that the transverse components of Poynting vector attenuate exponentially with distance. As usual, positive 〈S _φ〉 corresponds to counter-clockwise rotation when seeing against the beam propagation axis (the right-screw rule). In particular, within the plane of analysis z = 0, Eq. (21) can be reduced to the form

and 〈S _r〉 = 0. Furthermore, Eq. (23) can be evaluated approximately by making the Taylor expansion of 1/η (ω) in the vicinity of the beam’s central frequency. The average energy flow can be written in the form as the part of known results for non-dispersive media and the correction part from the dispersion

where Δω ³ << Δω and its higher order terms become small enough to be negligible. Thus, Eq. (24) can be reduced to

Obviously, Equation. (25) provides a general description of transverse energy flow for the considered dispersion of material and shows some remarkable difference from the propagation behavior in non-dispersive materials. It is worthwhile to consider two simple practical examples. In the example of LG_0,+1 + LG_0,-1, we can see from Fig. 2(a) that the direction of transverse energy flow presents anticlockwise spiral in vacuum, where l > 0 and 〈S _φ〉 = [Δωl/(2r|u|²)] > 0. However, when such a RHG beam propagates in the normal dispersion region of the LHM, η(ω ₀) is negative while ∂η/∂ω|_ω=ω0 is positive, it is obvious from Eq. (25) that its first term and its second term are both negative. Consequentially, 〈S _φ〉 < 0 and the direction of transverse energy flow is reversed in the normal dispersion region of the LHM due to the negative refractive index (see Fig. 2(a’)). In the other case of LG_{1, -1} +LG_{1, +1}, we can see from Fig. 2(b) that the direction of transverse energy flow exhibits clockwise spiral in vacuum, where l < 0 and 〈S _φ〉 = [Δωl/(2r |u|²)] < 0. Since in the normal dispersion region of the LHM, 〈S _φ〉 > 0, and the direction of transverse energy flow exhibits anticlockwise spiral (see Fig. 2(b’)).

Furthermore, the transverse energy flow of RHG beams is quite different in the anomalous dispersion region of the LHM, where both η (ω ₀) and ∂η/∂ω|_ω=ω0 are negative. Obviously, the first term of Eq. (25) is negative but its second term is positive. In this case, Eq. (25) proves that the direction of transverse energy flow could be affected significantly by the strength of the dispersion. As we see from Figs. 3(a) and 3(b), when (∂η/∂η)|_ω=ω0| < 2 |η (ω ₀)|/ω ₀, this point is evident by the fact that Δω/η (ω ₀)| > |ω ₀Δω (∂η/∂ω)|_ω=ω0/2η ² (ω ₀)|, which implies that the direction of transverse energy flow is determined by the negative refractive index of the LHM. Figs. 3(a’) and 3(b’) point out that since |(∂η/∂ω)|_ω=ω0| > 2 |η (ω ₀)|/ω ₀, the effect of the anomalous dispersion can reverse the direction of transverse energy flow, which is contrast the case in the normal dispersion region of the LHM. We can conclude that when a small amount of dispersion is introduced in the anomalous dispersion region of LHMs, the effect of dispersion is less than the effect of negative refractive index. With increased strength of the dispersion, one can also clearly observe that the effect of negative refractive index was counteracted by the effect of anomalous dispersion.

 

Fig. 2. Numerically computed field intensity distribution and transverse energy flow (green arrows) of RHG beams in vacuum (left) and in the normal dispersion region of the LHM (right), respectively, at z = 0. (a) and (a’): superposition LG_0,+1 + LG_0,-1, the transverse Poynting vector exhibits anticlockwise and clockwise spiral in vacuum and in the normal dispersion region of the LHM, respectively. (b) and (b’): superposition LG_1,-1 + LG_1,+1, the transverse Poynting vector presents clockwise and anticlockwise spiral in vacuum and in the normal dispersion region of the LHM, respectively.

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We now turn our attention to the angular momentum density and angular momentum flow of RHG beams in the dispersive LHM. The equation governing the momentum of an arbitrary subsystem generally take the form [31]

where momentum flow T, and momentum density G interact with other subsystems via force density F. It is necessary to include the dispersive characteristics of the material in describing the behavior of the electromagnetic wave. Thus the momentum conservation equations for the electromagnetic wave can be written in the form:

where I is 3 × 3 identity matrix. The electric and magnetic polarization vectors are given by P _e = ε ₀ (ε - 1)E and P _m = - μ ₀ (μ - 1)H, respectively. Bound electric current J = ∂P _e/∂t and bound electric charge ρ_e = ∇ · P _e have been accounted. Similarly, bound magnetic current M = ∂P _m/∂t and bound magnetic charge ρ_m = ∇ · P _m should be introduced to describe the angular momentum density in LHMs.

 

Fig. 3. Numerically computed field intensity distribution and transverse energy flow (green arrows) of RHG beams in the anomalous dispersion region of LHMs at z = 0. (a) and (a’): superposition LG₀₊₁ + LG_0-1, the transverse Poynting vector exhibits clockwise and anticlockwise spiral since |(∂η/∂ω)|_ω=ω0| > 2|η(ω ₀)|/ω ₀ and |(∂η/∂ω)|_ω=ω0| < 2|η(ω ₀)|/ω ₀ in (a) and (a’), respectively. (b) and (b’): superposition LG_{1, -1} + LG_{1, +1}, the transverse Poynting vector presents anticlockwise and clockwise spiral since | (∂η/∂ω)|_ω=ω0| < 2|η(ω ₀)|/ω ₀ and | (∂η/∂ω)|_ω=ω0| > 2|η(ω ₀)|/ω ₀ in (b) and (b’), respectively.

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The electric and magnetic field satisfies the relation |E| = c|B| /n. The momentum density can be reduced to

where e _s is the unit vector of Poynting vector. It is evident from Eq. (28) that the direction of momentum density is always antiparallel to the energy flow in lossless LHMs, where ∂ (nω)/∂ω is positive. Note that Eq. (28) is valid only for lossless medium, and its application to lossy media produces unphysical phenomena such as a negative energy in LHMs. In lossy media, the average momentum flux satisfies relation 〈T〉 = η 〈S〉/c [32]. While the sign of 〈T〉 follows exactly the sign of η, the momentum flow of RHG beams is opposite to the power flow in the region where η < 0. Similar to energy velocity, a momentum velocity may also be defined [33]

The energy velocity remains positive since it is in the region of η < 0. However, the momentum velocity becomes negative for part of this region. While the sign of 〈T〉 is negative, the momentum density 〈G〉 may be positive or negative in a frequency band with a negative index of refraction. Hence, the momentum density may be parallel or antiparallel to the transverse energy flow in dispersive LHMs. This is in contrast to the results for a lossless LHM, where the angular momentum density is always antiparallel to the energy flow. The result of Eq. (29) prove that: When η ² + κ ² + 2 (ω ₀ ηκ/γ_e) < 0, the angular momentum density is parallel to the transverse energy flow, while when η ² + κ ² + 2 (ω ₀ ηκ/γ_e) > 0, the angular momentum density is antiparallel to the transverse energy flow. Furthermore, as we see from Fig. 4, the angular momentum flow is always opposite to the transverse energy flow.

 

Fig. 4. Numerically computed field intensity distribution and angular momentum flow (green arrows) of RHG beams in the LHM for (∂η/∂ω)|_ω=ω0| > 2η(ω ₀)/ω ₀/ω ₀ (left) and (∂η/∂ω)|_ω=ω0 < 2η(ω ₀)/(ω ₀) (right). White arrow shows the direction of S _φ. (a) and (a’): superposition LG_{0, +1} +LG_0,-1. (b) and (b’): superposition LG_{1, -1} +LG_{1, +1}. Note that the angular momentum flow is always opposite to the transverse energy flow.

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5. Conclusion

We have investigated the propagation of rotating beams in dispersive LHMs on example of superposition of two LG modes with equal amplitude and different frequencies. From the Poynting vector equations for rotating beams, we find that the direction of transverse energy flow is reversed in the normal dispersion region of LHMs, due to the negative refractive index. In the anomalous dispersion region, however, the direction of transverse energy flow may be parallel or antiparallel to the case of vacuum, depending on the strength of dispersion. We have also found that although the transverse energy flow change their directions, the interference pattern rotation still remains unchanged. Finally, we have demonstrated that the angular momentum density may be parallel or antiparallel to the transverse energy flow in lossy LHMs, in sharp contrast to the results for a lossless LHM, where the angular momentum density is always antiparallel to the energy flow. In addition, the angular momentum flow is always opposite to the transverse energy flow. The investigation of propagation characteristics of rotating beams in LHMs may offer new fundamental insights into the nature of LHMs.